Siu [1]} proved that every complex K3 surface \(M\)  admits a Kähler metric and Yau [2]} proved that for every Kähler metric on \(M\)  there is a unique Ricci-flat Kähler metric in its cohomology class. These Ricci-flat metrics play the same role for K3 surfaces that constant curvature metrics play for Riemann surfaces.
If a finite group \(G\subset \operatorname{Diff}(M)\)  preserves some complex structure on \(M\)  then one can take a \(G\) -average of a Kähler metric and then apply Yau's theorem to find a \(G\) -invariant, Ricci-flat Kähler metric. Thus there are basically three Nielsen Realization problems for K3 surfaces, each stronger than the previous: realization by diffeomorphisms of \(M\) ; realization by isometries of some Ricci-flat metric on \(M\) ; and realization by complex automorphism of \(M\) .
